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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>For a particular solution, we let</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
{\bf x}^{(p)}={\bm \Psi}(t)\,{\bf u}(t).
\end{equation*}
</div>
<p class="continuation">Substituting it into (<a href="" class="xref" data-knowl="./knowl/eq8_10.html" title="Equation 6.6.6">(6.6.6)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
{\bm \Psi}(t) \, {\bf u}^{\prime}(t)={\bf g}(t) \to
\left(
\begin{array}{cc}
5 \cos t &amp; 5 \sin t\\
2 \cos t+\sin t &amp;-\cos t+2 \sin t
\end{array}
\right) \left(
\begin{array}{c}
u_1^{\prime}\\
u_2^{\prime}
\end{array}
\right)=\left(
\begin{array}{c}
-5 \cos t\\
\sin t
\end{array}
\right)
\end{equation*}
</div>
<p class="continuation">which further gives</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
u_1^{\prime}=-\cos 2 t+\sin 2 t,\quad
u_2^{\prime}=-2 \cos t (\cos t+\sin t).
\end{equation*}
</div>
<p class="continuation">Integration gives one set of solutions to <span class="process-math">\(u_1, u_2\)</span> as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
u_1=-\frac{1}{2} \cos 2t-\frac{1}{2} \sin 2 t,\quad
u_2=-t +\cos ^2 t-\frac{1}{2} \sin 2t.
\end{equation*}
</div>
<p class="continuation">The general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
\begin{aligned}
{\bf x}&amp;={\bm \Psi}(t)\,{\bf c}+{\bm \Psi}(t)\,{\bf u}(t)\\
&amp;= \left(-\frac{1}{2} \cos 2t-\frac{1}{2} \sin 2 t+C_1\right) \left(
\begin{array}{c}
5 \cos t\\
2 \cos t+\sin t
\end{array}
\right)+\left(-t +\cos ^2 t-\frac{1}{2} \sin 2t+C_2 \right) \left(
\begin{array}{c}
5 \sin t\\
-\cos t+2 \sin t
\end{array}
\right).
\end{aligned}
\end{equation*}
</div>
<span class="incontext"><a href="sec6_6.html#p-291" class="internal">in-context</a></span>
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